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Trigonometric Solutions of the Integro-differential Equation

  • William Alan Day
Chapter
  • 160 Downloads
Part of the Springer Tracts in Natural Philosophy book series (STPHI, volume 30)

Abstract

In this chapter we examine further the consequences† of the coupled and quasi-static approximation which leads to the integro-differential equation (2.1.3) for the temperature. We now abandon the boundary and initial value problem which was formulated in Theorem 2 and proceed to study a trigonometric solution of the integro-differential equation, that is to say a solution which has the form
$$ \theta (x,t) = {\rm Re} \sum {\Theta (x,\omega )\exp (i\omega t)} $$
(3.1.1)
where Θ is a (complex-valued) solution of the integro-differential equation
$$ \Theta ''(x,\omega ) = i\omega \left[ {(1 + a)\Theta (x,\omega ) - a\int_0^1 {\Theta (y,\omega )dy} } \right] $$
(3.1.2)
in which the primes denote derivatives with respect to x, and the sum (3.1.1) is taken over a finite set of distinct positive real exponents ω.

Keywords

Heat Flux Periodic Function Heat Equation BERNOULLI Number Total Energy Density 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1985

Authors and Affiliations

  • William Alan Day
    • 1
  1. 1.Mathematical InstituteUniversity of OxfordOxfordEngland

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